On the combinatorics of veering triangulations

在20世纪70年代和80年代，威廉·瑟斯顿彻底改变了低维拓扑学。他将许多新思想引入该领域，并进一步揭示了三流形组合研究与许多其他领域之间令人兴奋的联系：Teichmuller理论、全纯动力学、Kleinian群、几何群论等等。他开创的新技术之一是使用理想三角形来表示双曲三流形。这使得拓扑学家一方面可以使用代数几何的工具，另一方面，也可以为一堆杂乱无章的例子找到一个组织原则。理想三角剖分的概念现在已经被推广和专门化了很多次，给出了角度三角剖分[Casson]，紧绷理想三角剖分[Lackenby]和转向三角剖分[Agol， HRST]。Francois Gueritaud发现了纤维流形的Cannon-Thurston图和它们的转向三角形之间的重要联系。亚伊尔·明斯基和塞缪尔·泰勒给出了地下投影机制和转向三角测量结构之间的有趣关系。该项目的目标是了解在非光纤环境下，转向三角测量的组合如何告知相应的Cannon-Thurston地图和三管管通用覆盖的“纤维结构”。

In the 1970's and 1980's William Thurston revolutionised low-dimensional topology. He introduced many new ideas into the field, and furthermore revealed exciting connections between the combinatorial study of three-manifolds and many other areas:Teichmuller theory, holomorphic dynamics, Kleinian groups, geometric group theory, and more.One of the new techniques he pioneered was the use of ideal triangulations to represent hyperbolic three-manifolds. This allowed topologists, on the one hand, access to tools in algebraic geometry and, on the other hand, an organising principle for what had been a hodge-podge of disparate examples. The idea of ideal triangulations has now been generalised and specialised many times to give angled triangulations [Casson], taut ideal triangulations [Lackenby], and veering triangulations [Agol, HRST].Francois Gueritaud has found an important connection between the Cannon-Thurston map for fibered manifolds and their veering triangulations. Yair Minsky and Samuel Taylor have given an interesting relation between the machinary of subsurface projections and the structure of veering triangulations. The goal of this project is to understand how the combinatorics of a veering triangulation, in the non-fibered setting, inform the cooresponding Cannon-Thurston map and the "fibered structure" of the universal cover of the three-manifold.